4. Suppose we are working with a circular dart board with 6 concentric regions labelled

1 (outermost) to 6 (disc at the center). Assume that the dart board has radius r and

that a dart thrown at the board always hits the board.

It is easy to see that the probability of scoring i points on a single throw is

(a) Show that

P(scoring i points) =

Area of region i

Area of the dart board

P(scoring i points) =

(7-i)²-(6-1)²

6²

Note that this probability is independent of the radius of the dartboard, why?

(b) Show that function constructed above satisfies the probability axioms.

(c) Show that the probability function constructed above is an increasing function

of i.

5. Suppose P is a probability function on the pair (sample space, sigma algebra): (S,B).

Using only the three defining properties of P, and for any A, B € B, show that:

(a) P(A) = 1- P(A).

(b) P(AUB) = P(A) + P(B) - P(ANB).

(c) If A C B, then P(A) ≤ P(B).